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The simplicial boundary of a CAT(0) cube complex

Mark F Hagen

Algebraic & Geometric Topology 13 (2013) 1299–1367

For a CAT(0) cube complex X, we define a simplicial flag complex X, called the simplicial boundary, which is a natural setting for studying nonhyperbolic behavior of X. We compare X to the Roller, visual and Tits boundaries of X, give conditions under which the natural CAT(1) metric on X makes it isometric to the Tits boundary, and prove a more general statement relating the simplicial and Tits boundaries. The simplicial boundary X allows us to interpolate between studying geodesic rays in X and the geometry of its contact graph ΓX, which is known to be quasi-isometric to a tree, and we characterize essential cube complexes for which the contact graph is bounded. Using related techniques, we study divergence of combinatorial geodesics in X using X. Finally, we rephrase the rank-rigidity theorem of Caprace and Sageev in terms of group actions on ΓX and X and state characterizations of cubulated groups with linear divergence in terms of ΓX and X.

CAT(0) cube complex, contact graph, divergence, rank-one isometry, simplicial boundary
Mathematical Subject Classification 2010
Primary: 05C25, 20F65, 57M99
Received: 4 January 2012
Revised: 1 May 2012
Accepted: 20 December 2012
Published: 24 April 2013
Mark F Hagen
Department of Mathematics
University of Michigan
530 Church Street
1859 East Hall
Ann Arbor, MI 48109